Question

The equation for standing wave in a stretched string is given by $y=5\sin \left( \dfrac{\pi x}{3} \right)\cos \left( 40\pi t \right)$, where $x$ and $y$ are in cm and $t$ in second. The separation between two consecutive nodes is (in cm):
A.) 1.5
B.) 3
C.) 6
D.) 4

Answer 1

Hint: On a standing wave there are points which have zero vertical displacement always. These points are known as nodes. To find out the nodes form the equation of a standing wave, the displacement should be equated to zero for all time values. The values of the horizontal position, which will satisfy this criterion, will give the position of the nodes.
Formula used:
For $\sin \theta =0$,
$\theta =n\pi $, where n is an integer.

Complete step by step answer:
A standing wave has points on it that have zero vertical displacement at all times. These points are known as nodes. These are essentially the points where maximum destructive interference occurs and hence, the point virtually has no displacement.
The general solution for the position of nodes can be found out from the equation of the standing wave by considering the displacement to be zero at all times and calculating the horizontal positions on the wave which satisfy this criterion.
Hence, let us do that.

The given standing wave equation is,

$y=5\sin \left( \dfrac{\pi x}{3} \right)\cos \left( 40\pi t \right)$
We will consider$\cos \left( 40\pi t \right)$for making the displacement zero, since we have to find out the position on the wave for which $y$ is zero regardless of the time $t$.
Hence, now putting $y=0$, we get,
$0=5\sin \left( \dfrac{\pi x}{3} \right)\cos \left( 40\pi t \right)$
Not considering $\cos \left( 40\pi t \right)=0$ as explained above, we get,
$\sin \left( \dfrac{\pi x}{3} \right)$ --(1)

Now, for $\sin \theta =0$,
$\theta =n\pi $, where n is an integer.
Using this in (1), we get,
$\dfrac{\pi x}{3}=n\pi $
$\therefore x=3n$--(2)

where n is an integer.

Hence, nodes are found at $x=3n$ for $n=1,2,3....$

Therefore, the distance between two nodes is given by, the difference in the positions of two consecutive nodes at ${{x}_{n}}$ and ${{x}_{n+1}}$

Therefore, using (2), we get,

${{x}_{n+1}}-{{x}_{n}}=3\left( n+1 \right)-3n=3n+3-3n=3cm$
Hence, the distance between two consecutive nodes is $3cm$.

Therefore, the correct option is B) $3cm$.

Note: Students often get confused between nodes and antinodes and which one actually corresponds to the condition of zero displacement on the standing wave and which one for maximum displacement. An easy way to remember, is to relate the word ‘node’ with ‘none’ (both of them are similar in their spellings) implying that the displacement at a ‘node’ is ‘none’ (zero) and antinodes obviously being the opposite have maximum displacement (anti-none).